Linear Equations

Learning Objectives

  • Understand linear equations and their various forms (Slope-Intercept, Point-Slope, Standard).
  • Calculate distance and the midpoint between two points on the Cartesian plane.
  • Define and compute the slope of a line.
  • Identify parallel and perpendicular lines based on their slopes.

A linear equation represents a straight line on the Cartesian coordinate plane. It is the foundation for modeling constant rates of change and solving problems involving speed, cost, and mixtures.

Linear Equation

An algebraic equation in which each term is either a constant or the product of a constant and a single variable, representing a straight line when graphed.

Forms of Linear Equations

Depending on the information available, linear equations can be expressed in different forms.

  • Slope-Intercept Form: y=mx+by = mx + b. Identifies slope mm and y-intercept bb.
  • Point-Slope Form: y−y1=m(x−x1)y - y_1 = m(x - x_1). Used when a point (x1,y1)(x_1, y_1) and slope mm are known.
  • Standard Form: Ax+By=CAx + By = C. Useful for finding intercepts where A≥0A \ge 0.

Interactive Simulation

Use the interactive converter below to explore how linear equations switch between Point-Slope form, Slope-Intercept form, and Standard form. Drag the coordinates or adjust the sliders to see the transformations in real-time.

Linear Equation Forms Converter

Point-Slope Coordinates
Point 1: P1(x1,y1)P_1(x_1, y_1)
Point 2: P2(x2,y2)P_2(x_2, y_2)
Active Conversions
Point-Slope
y−(−3)=1.00(x−(−2))y - (-3) = 1.00(x - (-2))
Slope-Intercept
y=1.00x−1.00y = 1.00x - 1.00
Standard Form (Ax + By = C)
(5.0)x−(5.0)y=5.0(5.0)x - (5.0)y = 5.0
Point 1Point 2
Rise / Run5 / 5
Slope (mm)1.00
y-Intercept (bb)-1.00

Midpoint

The exact center point dividing a line segment into two equal halves.

Distance and Midpoint

Given two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) on a Cartesian plane, we can find the distance between them and their exact center.

Distance Formula

Calculates the straight-line distance between two points (derived from the Pythagorean theorem).

d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

Variables

SymbolDescriptionUnit
ddThe distance between the two pointsunits
(x1,y1)(x_1, y_1)Coordinates of the first point-
(x2,y2)(x_2, y_2)Coordinates of the second point-

Midpoint Formula

Calculates the exact center coordinate between two points.

M=(x1+x22,y1+y22)M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)

Variables

SymbolDescriptionUnit
MMThe midpoint coordinates (x_m, y_m)-
(x1,y1)(x_1, y_1)Coordinates of the first point-
(x2,y2)(x_2, y_2)Coordinates of the second point-

Slope

A measure of the steepness and direction of a line, calculated as the ratio of the change in the y-axis to the change in the x-axis ("Rise over Run").

Understanding Slope

The slope mm represents the "Rise over Run", indicating the steepness and direction of the line.

  • Positive Slope: Line rises from left to right.
  • Negative Slope: Line falls from left to right.
  • Zero Slope: Horizontal line (y=by = b).
  • Undefined Slope: Vertical line (x=ax = a).

Slope Formula

Calculates the slope (steepness) of a line given two points.

m=ΔyΔx=y2−y1x2−x1m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}

Variables

SymbolDescriptionUnit
mmThe slope of the line-
Δy\Delta yThe change in the y-coordinates (rise)-
Δx\Delta xThe change in the x-coordinates (run)-
(x1,y1)(x_1, y_1)Coordinates of the first point-
(x2,y2)(x_2, y_2)Coordinates of the second point-

Interactive Visualizer

Interactive Simulation

Experiment with the slope mm and y-intercept bb below. Observe how changing mm rotates the line and changing bb shifts it vertically.

Linear Equation Explorer

Current form
y=xy = x
Converted meaning
y=xy = x

Standard form converts by solving Ax+By=CAx + By = C for yy when Bneq0B \\neq 0. If B=0B = 0, the graph is vertical and the slope is undefined.

y = x. Line with slope 1 and y-intercept 0.

xy
State

Line with slope 1 and y-intercept 0

Slope

1

Intercept cue

y-intercept 0

Parallel and Perpendicular Lines

Slopes reveal the geometric relationship between two lines.

  • Parallel Lines: Have equal slopes (m1=m2m_1 = m_2). They never intersect.
  • Perpendicular Lines: Have slopes that are negative reciprocals (m1⋅m2=−1m_1 \cdot m_2 = -1).

Interactive Simulation

Interactive Simulation

Below is an interactive calculator to explore the properties of y=mx+by = mx + b. You can change the slope mm and y-intercept bb to see how the line changes in real-time.

Slope-Intercept Explorer ($y = mx + b$)

y=1.5x+1y = 1.5x + 1
Slope m (steepness & direction)1.5
Y-Intercept b (vertical shift)1

Interactive Insights

Slope ($m = 1.5$): Line rises from left to right.

Y-Intercept ($b = 1$): The line intersects the vertical y-axis exactly at point $(0, 1)$.

Slope=RiseRun=3.02=1.5\text{Slope} = \frac{\text{Rise}}{\text{Run}} = \frac{3.0}{2} = 1.5
xyRun: 2Rise: 3.0(0, 1)
Y-Intercept (0, b)
Run (Horizontal)
Rise (Vertical)
Key Takeaways
  • Slope-Intercept Form: y=mx+by=mx+b is the most useful for graphing and understanding the "starting value" bb.
  • Special Slopes: Horizontal lines have slope 00 (y=by=b); Vertical lines have an undefined slope (x=ax=a).
  • Parallel vs Perpendicular: Parallel lines have the same slope; Perpendicular lines have negative reciprocal slopes.
  • Golden Rule: When solving linear inequalities, flip the inequality sign if multiplying or dividing by a negative number.
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